40th astin colloquim madrid, 19-22 june 2011 · 2018-05-22 · pirra, forte, ialenti – 40th astin...

Implementing a Solvency II internal model:

Bayesian stochastic reserving and parameter estimation

40th ASTIN Colloquim

Madrid, 19-22 June 2011

Marco [email protected]

Salvatore [email protected]

Matteo [email protected]

Pirra, Forte, Ialenti – 40th ASTIN Colloquium – Madrid, June 19-22, 2011 2

Agenda

1. Solvency II

2. Bayesian reserving methods

3. Bayesian Fisher Lange (BFL)

4. Case Study : Stability, Sensitivity, Backtesting.

5. Final remarks

Pirra, Forte, Ialenti – 40th ASTIN Colloquium – Madrid, June 19-22, 2011

Solvency II is the new insurance directive that becomes effective on 1January 2013.

Under Solvency II capital requirements will be determined on the basisof the risk profile of insurance companies and the way companiesmanage such risks, therefore providing the right incentives for soundrisk management practices and enhanced transparency.

Under Solvency II, the valuation of technical provisions follows the transfer value principle, underwhich the value of technical provisions shall correspond to the current amount the insurer would haveto pay if was to transfer its insurance obligations immediately to another insurer.

To achieve a valuation consistent with this principle, the technical provisions are calculated as a bestestimate plus a risk margin.

I. The best estimate corresponds to the probability-weighted average of future cash-flows, takingaccount of the time value of money.

II. The risk margin represents the cost of providing an amount of eligible own funds equal to theSolvency Capital Requirement necessary to support the insurance and reinsurance obligationsover the lifetime thereof.

1 – Solvency II

3


The Solvency Capital Requirement (SCR) is the risk-based capitalrequirement for undertakings under Solvency II. It is calibrated to a99.5% Value at Risk confidence level over one year.

In structure the SCR is composed of a number of “modules” which inturn are composed of “sub-modules”. The capital requirements arisingfrom these sub-modules and modules are aggregated using a correlationmatrix.

In the Solvency II regime the Solvency Capital Requirement (SCR) is tobe calculated by undertakings in accordance with the standard formulaor using a full internal model, or using a combination of both a partialinternal model and the standard formula.

Internal models: the validation process is crucial (”Validation tool”means any approach designed to gain comfort that the internal model isappropriate and reliable. Some common examples include: back testing;sensitivity testing; stress and scenario testing; profit and loss attribution;benchmarking; reserve stress test)

1 – Solvency II

4


EIOPA Report on the fifth Quantitative Impact Study (QIS5) forSolvency II, Non-Life Underwriting Risk Composition:

1 – Solvency II

5

The paper focuses on the Reserve risk (which results from fluctuationsin the timing and amount of claim settlements).


Insurance is, by nature, a very uncertain subject. Insured events occur atrandom times and, particularly in the field of general insurance, theamounts of the claims are also random.

In the general insurance market, insurers need to use the data gatheredfrom previous years of experience to make predictions about futureliabilities.

At the heart of classical non-Bayesian statistical analysis is the conceptof asymptotically, which suggests that a large enough number ofobservations from the same phenomenon will produce certain statisticswith properties close to some convenient models; the more observations,generally, the closer the results will be to the ideal.

The concept works well for repeatable experiments, such as the toss of acoin, but it requires a leap of faith when only a relatively small numberof observations are available for an ever-changing environment.

2 – Bayesian Methods

6


Bayesian statistics makes use of two sources of information: theobserved data (called the “likelihood”), and additional externalinformation that may not necessarily be present in the observed data(this is called the “prior”). Both the likelihood and the prior areformulated as probability distributions. Bayesian statistics is in contrastto classical statistics, which is generally limited to using the observeddata only.

There are a number of benefits that Bayesian stochastic reservingmodels bring. The first benefit is that Bayesian modelling is flexibleenough to build models that are similar to currently used reservingmodels.

Bayesian models provide a formal framework for integrating actuarialjudgment where an actuary does not consider the pure data alone tocompletely describe all of the information relevant to valuing theliabilities.


7


In an actuarial context, the prior distribution allows you to incorporateinformation that is not in the data. Actuaries will already be familiarwith this idea, as a significant part of any reserving exercise is theapplication of actuarial judgment.

Thus, from a Bayesian point of view, the prior model is a powerful wayfor past experience to be brought to bear on a current problem.

The impact on the final results of introducing a prior distributiondepends on a number of factors.

The results available are useful to understand the overall distribution ofreserves, but also the distribution of each future payment (and thuspotential values for each future payment). Having the full distributionof results makes it possible to use any percentile on the distribution.


8


n

ink

ki

ji

iniinijiRateNRNP

1

,

,

1,1,,

9

3 – Bayesian Fisher-Lange (BFL)

The deterministic Fisher-Lange model (“average cost per claim method”)determines the future payments (and the claim reserve as an obviousconsequence), by multiplying the estimated number of future claims (NPij) andthe corresponding expected average cost (ACij) :

assuming:

claims to be settledproportion of claim

settled

settlement speed

(2)

(1)1

,,,

njiiriACNPY

jijiji


3 – Bayesian Fisher-Lange (BFL)

The distributions used in the model refer to the following variables:

1. proportion of claims settled:

2. settlement speed:

3. average cost per claim:

Posterior distributions of the parameters and predictive distribution of the model builtup through Monte Carlo Markov Chain (MCMC) techniques.

RateRate

jji NRate ,~,

jjji N ,~,

AC

j

AC

jjiAC ,~,

RateRate

j

Rate

j N .,.~

jjj N .,.~

AC

j

AC

j

AC

j N .,.~


4 – Case Study: General Liability

The initial data set is represented by the run-off triangle of incremental

payments of an insurance company operating in the LoB third party liability


4 – Case Study: BFL Outstanding Claim Reserve

The discounted values are obtained considering the Interest Rate TermStructure 2007 given by CEIOPS in QIS Technical Specifications. Theinterest rates do not consider any illiquidity premium.

(Euro Thousands) (Euro Thousands)


4 – Case Study: BFL vs. ODP (Discounted)

As far as the variability is concerned the BFL leads to higher values as it has adouble level of stochasticity. The ODP has a very high variability on oldergenerations and lower values on recent generations (typical characteristic of theODP model and in general of the chain ladder methods). The BFL giveshomogeneous values on different generations due to its bayesian nature and

construction.

(Euro Thousands) (Euro Thousands)


4 – Case Study: Parameter θjrate OCR Stability

The results show a good stability of the model, both in terms of expectedvalue and standard deviation of the outstanding claim reserve: even ifthe input parameter in the prior distribution changes significantly theexpected value remains pretty much the same.

1. proportion of claims settled: RateRate

j

Rate

j N .,.~


4 – Case Study: Parameter θjv OCR Stability

The results show a good stability of the model, both in terms of expectedvalue and standard deviation of the outstanding claim reserve: even ifthe input parameter in the prior distribution changes significantly theexpected value remains pretty much the same.

2. settlement speed:jjj N .,.~


4 – Case Study: Parameter θjAC OCR Stability

The results show a good stability of the model both in terms of expected value andstandard deviation of the outstanding claim reserve. Although the differences seemrelevant, the model leads to a good convergence to the posterior distribution: evenreducing the prior average costs of a percentage equal to 50% the reduction of the expectedvalue of the outstanding claim reserve is limited to a 12%.

3. average cost per claim: AC

j

AC

j

AC

j N .,.~


4 – Case Study: Prior vs. Posterior Distribution Rateij

The comparison of the figures demonstrates the optimal convergence ofthe model to the posterior distributions and confirms the considerationsoutlined in the stability analyses: for example the Rate12,1 passes from aprior expected value equal to 100% to a posterior expected value equal to91.71%.


4 – Case Study: Prior vs. Posterior Distribution vij

The comparison of the figures demonstrates the optimal convergence ofthe model to the posterior distributions and confirms the considerationsoutlined in the stability analyses: for example the v12,2 passes from aprior expected value equal to 75% to a posterior expected value equal to80.1%.


4 – Case Study: Prior vs. Posterior Distribution ACij

The comparison of the figures demonstrates the optimal convergence ofthe model to the posterior distributions and confirms the considerationsoutlined in the stability analyses: for example the AC12,9 passes from aprior expected value equal to 25k to a posterior expected value equal to29k.


4 – Case Study: Backtesting

The approach followed in the work in order to understand if the reservepredicted by the model matches the reserve held by the insurancecompany is to compare prior year development to model predictions,that is to say compare the probability distribution and the expectedvalue of the first diagonal of the run-off triangle obtained by theexclusion of the last generation and the actual paid value written in thebalance sheet.

Y[i,j]Underwriting Year 0 1 2 3 4 5 6 7 8 9 10 11

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Development Year


4 – Case Study: Backtesting

The actual paid value in the case studyis 118,438k.


4 – Case Study: OCR Sensitivities

1. proportion of claims settled: RateRate

jji NRate ,~,

2. settlement speed:jjji N ,~,


4 – Case Study: Solvency II Results

Internal 1Y Merz-Wuthrich ODP Bootstrap Bayesian Fisher-Lange

Best Estimate 256,585€ 256,653€ 266,200€ Risk Margin (% BE) 1.01% 1.20% 2.04%

Reserve Risk Capital (% BE) 6.30% 7.94% 14.13%σ (1 year) 2.75% 3.44% 6.04%

QIS5 Market Wide Merz-Wuthrich ODP Bootstrap Bayesian Fisher-Lange




4 – Case Study: Solvency II vs. QIS5 Final Report EIOPA

Internal 1Y Merz-Wuthrich ODP Bootstrap Bayesian Fisher-Lange




Conclusions

The prior distribution allows to incorporate information that is not inthe data. Actuaries will already be familiar with this idea, as asignificant part of any reserving exercise is the application of actuarialjudgment.

The bayesian methodology combined with the Fisher Lange model hasthe advantage of explicitly taking into account the settlement andreserving policies of the insurer.

The impact on the final results of introducing a prior distribution can beadequately explained.

The variability measure (sigma) and the reserve risk capital aresignificantly affected by the probabilistic structure of the model and bythe insurer dimensions.


Further discussion – see paper

The choice of the internal model for the reserve risk assessment has agreat importance; that is the reason a set of validation criteria should bedefined and verified through a backtesting analysis.

The most appropriate backtesting methodology and the choice amongdifferent stochastic reserving methods are still being debated. Othervalidation tools?

QIS5: standard formula limits and alternative solutions.


References

[1] AISAM-ACME, 2007, Study on non-life long tail liabilities. Reserve risk and risk margin assessmentunder Solvency II. Available on www.amice-eu.org.

[2] CEIOPS, 2010,QIS5 Technical Specifications. Available on www.ceiops.org.[3] Christofides S., 1990, Regression Models based on log-incremental payments. Claims Reserving

Manual,Vol 2, Institute of Actuaries.[4] England P., Verrall R, 2002, Stochastic Claims Reserving in General Insurance. British Actuarial

Journal 8, III, 443-544.

[5] England P., Verrall R., 2006, Predictive Distributions of Outstanding Liabilities in GeneralInsurance. Annals of Actuarial Science: 1, 221-270.

[6] European Commission, 2009, Solvency II directive (2009/138/EC). Available on ec.europa.eu.

[7] Fisher, W., Lange J., 1973, Loss Reserve Testing: A Report Year Approach, Casualty Actuarial SocietyProceedings, Casualty Actuarial Society.

[8] Forte S., Ialenti M., Pirra M., 2008, Bayesian Internal Models for the Reserve Risk Assessment,Giornale dell’ Istituto Italiano degli Attuari, Volume LXXI N.1, 39-58.

[9] Forte S., Ialenti M., Pirra M., 2010, A reserve risk model for a non-life insurance company.Mathematical and Statistical Methods for Actuarial Sciences and Finance. Springer.

[10] IAIS, 2007, Guidance paper on the structure of regulatory capital requirements, Fort Lauderdale.

[11] International Actuarial Association IAA, 2009, Measurement of liabilities for insurance contracts:current estimates and risk margins. Available on www.actuaries.org.

[12] Klugman S. A., Panjer H. H., Willmot G. E., 2008, Loss Models: From Data to Decisions, 3rd Edition,Wiley and Sons Edition.


References

[13] Li J., 2006, Comparison of Stochastic Reserving Methods. Australian Actuarial Journal, volume 12,number 4.

[14] Mack T., 1993, Distribution-free Calculation of the Standard Error of Chain Ladder ReserveEstimates. ASTIN Bulletin 23, 214-225.

[15] Merz M., Wuthrich M., 2008, Modelling The Claims Development Result For Solvency Purposes.Casualty Actuarial Society E-Forum.

[16] Meyers G., 2007, Thinking Outside the Triangle. Paper presented to the 37th ASTIN Colloquium,Florida.

[17] Meyers G., 2007, Estimating Predictive Distributions for Loss Reserve Models. Variance volume 1,issue 2.

[18] Meyers G., 2009, Stochastic Loss Reserving with the Collective Risk Model. Variance, volume 3,issue 2.

[19] Ntzoufras I., Dellaportas P., 2002, Bayesian modelling of outstanding liabilities incorporating claimcount uncertainty. North American Actuarial Journal, volume 6 issue 1.

[20] Rebonato R., 2007, Plight of the Fortune Tellers: Why We Need to Manage Financial Risk Differently.Princeton University Press.

[21] Scollnik D. P. M., 2004, Bayesian Reserving Models Inspired by Chain Ladder methods andimplemented using WINBUGS, Actuarial Research Clearing House 2004 issue 2.

[22] Verrall R., 2004, A Bayesian Generalized Linear Model for the Bornhuetter Ferguson Method ofClaims Reserving. North American Actuarial Journal, volume 8, number 3.

[23] Verrall R., 2007, Obtaining Predictive Distributions for Reserves Which Incorporate Expert Opinion.Variance, volume 1, issue 1.

[24] Wuthrich M.V., Buhlmann H., Furrer H., 2008,Market-Consistent Actuarial Valuation. Springer.


Thank You !Marco Pirra

[email protected]

Salvatore [email protected]

Matteo [email protected]

40th astin colloquim madrid, 19-22 june 2011 · 2018-05-22 · pirra, forte, ialenti – 40th astin...

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